Problem: Factor the following expression: $-9$ $x^2$ $-29$ $x$ $-6$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-9)}{(-6)} &=& 54 \\ {a} + {b} &=& & & {-29} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $54$ and add them together. The factors that add up to ${-29}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-2}$ and ${b}$ is ${-27}$ $ \begin{eqnarray} {ab} &=& ({-2})({-27}) &=& 54 \\ {a} + {b} &=& {-2} + {-27} &=& -29 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-9}x^2 {-2}x {-27}x {-6} $ Group the terms so that there is a common factor in each group: $ ({-9}x^2 {-2}x) + ({-27}x {-6}) $ Factor out the common factors: $ x(-9x - 2) + 3(-9x - 2) $ Notice how $(-9x - 2)$ has become a common factor. Factor this out to find the answer. $(-9x - 2)(x + 3)$